A typical UK GCSE Exam Question

A typical question involves several intermediate constructions followed by a question on how to complete the same compositions of transformations using just one transformation instead

Angle between axes

Axes angle

1st Reflection

2nd Reflection

Method 1

Method 2

Single Transformation 1

Single Transformation 2

1st Reflection

2nd Reflection

A = (4.034715013002978, 3.392397200445023)
B = (4.2024798106460235, 6.939424350612247)
C = (6.479287778658758, 6.939424350612247)
D = (6.671018975965094, 5.501440370814719)
E = (5.161135797177703, 5.261776374181799)
F = (5.328900594820755, 3.2965316017918536)
poly1, a, b, c, d, e, f = Polygon(A, B, C, D, E, F)
G = (-15, 9)
g = Circle(G, 1)
H = (-15, 5)
h = Circle(H, 1)
i = OrthogonalLine(G, yAxis)
j = OrthogonalLine(H, yAxis)
I = Intersect(g, i, 1)
J = Intersect(h, j, 1)
K = Point(g)
L = Point(h)
k = Segment(G, I)
l = Segment(G, K)
m = Segment(H, L)
n = Segment(H, J)
α = Angle(I, G, K)
β = Angle(J, H, L)
u = false
v = false
M = Intersect(xAxis, yAxis)
N = Point(xAxis)
N' = Rotate(N, α, M)
p = Line(M, N')
N'' = Rotate(N', β, M)
q = Line(M, N'')
A' = Mirror(A, q)
B' = Mirror(B, q)
C' = Mirror(C, q)
D' = Mirror(D, q)
E' = Mirror(E, q)
F' = Mirror(F, q)
poly1', a', b', c', d', e', f' = Polygon(A', B', C', D', E', F')
A'' = Mirror(A', p)
B'' = Mirror(B', p)
C'' = Mirror(C', p)
D'' = Mirror(D', p)
E'' = Mirror(E', p)
F'' = Mirror(F', p)
poly1'', a'', b'', c'', d'', e'', f'' = Polygon(A'', B'', C'', D'', E'', F'')
r = CircleSector(M, N, N')
s = CircleSector(M, N', N'')
o = false
t = false
w = false
z = false
a_1 = false
b_1 = false
c_1 = false
A'_1 = Mirror(A, p)
B'_1 = Mirror(B, p)
C'_1 = Mirror(C, p)
D'_1 = Mirror(D, p)
E'_1 = Mirror(E, p)
F'_1 = Mirror(F, p)
poly1'_1, a'_1, b'_1, c'_1, d'_1, e'_1, f'_1 = Polygon(A'_1, B'_1, C'_1, D'_1, E'_1, F'_1)
O = (-16, -3)
e_1 = Line(O, xAxis)
f_1 = Circle(O, 1)
P, Q = Intersect(f_1, e_1)
g_1 = Semicircle(P, Q)
R = Point(g_1)
γ = Angle(Q, O, R)
h_1 = Segment(O, Q)
i_1 = Segment(O, R)
A'_2 = Rotate(A, γ, M)
B'_2 = Rotate(B, γ, M)
C'_2 = Rotate(C, γ, M)
D'_2 = Rotate(D, γ, M)
E'_2 = Rotate(E, γ, M)
F'_2 = Rotate(F, γ, M)
poly1'_2, a'_2, b'_2, c'_2, d'_2, e'_2, f'_2 = Polygon(A'_2, B'_2, C'_2, D'_2, E'_2, F'_2)
j_1 = Segment(M, A'_2)
k_1 = Segment(M, A)
ScaleFactor = 1 (slider from -1 to 1, step 0.01)
A'_3 = Dilate(A, ScaleFactor, M)
B'_3 = Dilate(B, ScaleFactor, M)
C'_3 = Dilate(C, ScaleFactor, M)
D'_3 = Dilate(D, ScaleFactor, M)
E'_3 = Dilate(E, ScaleFactor, M)
F'_3 = Dilate(F, ScaleFactor, M)
poly1'_3, a'_3, b'_3, c'_3, d'_3, e'_3, f'_3 = Polygon(A'_3, B'_3, C'_3, D'_3, E'_3, F'_3)
l_1 = Segment(B'', B)
m_1 = Segment(F'', F)
p_1 = CircleArc(M, A, A'_2)
q_1 = CircleArc(M, C, C'_2)
S = Mirror(A'_1, q)
T = Mirror(B'_1, q)
U = Mirror(C'_1, q)
V = Mirror(D'_1, q)
W = Mirror(E'_1, q)
Z = Mirror(F'_1, q)
poly2, s_1, t_1, u_1, v_1, w_1, z_1 = Polygon(S, T, U, V, W, Z)
d_1 = false

Tool: Complex Multiplication (command ComplexMultiplication)
Select Points at O(0, 0), P(0, -1), A and B for A*B
Input: O, P, A, B
Output: j, AB

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
b = Segment(O, A)
f = Segment(O, B)
G = Intersect(d, xAxis, 1)
g = Circle(D, P, G)
E = Intersect(g, yAxis, 2)
e = Circle(O, E)
a, h = AngularBisector(f, b)
D' = Mirror(D, a)
i = Ray(O, D')
AB = Intersect(e, i, 1)
j = Segment(O, AB)

Tool: Complex Division (command ComplexDivision)
Select Points O(0, 0), P(0, -1), A and B for A/B
Input: O, P, A, B
Output: l, C

A/B

c = Circle(O, A)
d = Circle(O, B)
D = Intersect(c, xAxis, 2)
e = AngularBisector(A, O, D)
B' = Mirror(B, e)
f = Ray(O, B')
E = Intersect(c, yAxis, 1)
F = Intersect(d, xAxis, 1)
g = Segment(F, E)
G = Midpoint(g)
h = OrthogonalLine(G, g)
i = Segment(P, F)
H = Midpoint(i)
j = OrthogonalLine(H, i)
I = Intersect(h, j)
k = Circle(I, F)
J = Intersect(k, xAxis, 2)
p = Circle(O, J)
C = Intersect(p, f)
l = Segment(O, C)

Tool: Complex Addition (command ComplexAddition)
Select Points O(0, 0), A and B for A + B
Input: O, A, B
Output: C', c

A + B

u = Vector(O, B)
C' = Translate(A, u)
c = Segment(O, C')

Tool: Complex Subtraction (command ComplexSubtraction)
Select Points O(0, 0), A and B for A - B
Input: O, A, B
Output: c, A'

A - B

u = Vector(B, O)
A'_1 = Translate(A, u)
A' = Translate(A, u)
c = Segment(O, A')